How to find equations of a sub-Riemannian problem

Question

I am working on sub-Riemannian geometry and try to understand what are the tools to find the equations of a sub-Riemannian problem. Here is an example:

Let us consider the system defined by a lagrangian:

\begin{equation} L=\frac{1}{2}m(\overset{\cdot}{x}^2+\overset{\cdot}{y}^2)+\frac{1}{2}I\overset{\cdot}{\theta}^2 \end{equation}

And take the following non-holonomic constraint

\begin{equation} \mathscr{D}=span\left\{\cos\theta\frac{\partial}{\partial x}+\sin\theta\frac{\partial}{\partial y},\frac{\partial}{\partial\theta}\right\} \end{equation} So we have, \begin{equation} \mathscr{D}^\perp=span\{\omega\} \end{equation} with $\omega=-\sin\theta dx+\cos\theta dy$

If I understand clearly the sub-Riemannian problem is to minimize \begin{equation} \int_0^TLdt \end{equation} under the constaint $\dot{z}(t)\in\mathscr{D}(z(t))$. Now, how do I find these equations? So far I am not asking how to solve these equations but just to understand how to find them in the first place.

Thanks in advance for your help.

Edit:

I quite understand your answer, thank you for the clear explanation. I am though not really familiar with canonical coordinates. I agree with the expression of the Hamiltonian.

Now, I am right if I say that the Hamilton's equations are given by:

\begin{equation} \begin{cases} \overset{\cdot}{x}(t)&=\frac{\partial H}{\partial p_x}=p_x\cos^2\theta +p_y\cos\theta\cdot\sin\theta \\ \overset{\cdot}{y}(t)&=\frac{\partial H}{\partial p_y}=p_y\sin^2\theta +p_x\cos\theta\cdot\sin\theta \\ \overset{\cdot}{\theta}(t)&=\frac{\partial H}{\partial p_\theta}=p_\theta\\ \overset{\cdot}{p_x}(t)&=-\frac{\partial H}{\partial x}= 0 \\ \overset{\cdot}{p_y}(t)&=-\frac{\partial H}{\partial y}= 0 \\ \overset{\cdot}{p_\theta}(t)&=-\frac{\partial H}{\partial \theta}= (p_x^2-p_y^2)\sin(\theta)cos(\theta)+p_xp_y(\sin^2(\theta)-\cos^2(\theta)) \\ \end{cases} \end{equation}

I'm not sure to understand how to interpret them. Are these the equations of the sub-Riemannian problem?

Answer 1

You can reformulate your problem in the language of geometric control (where your dynamical system is sometimes called Dubin's car). Let $$ X_1=\frac{1}{\sqrt{m}}(\cos\theta\partial_x + \sin\theta\partial_y),\qquad X_2=\frac{1}{\sqrt{I}}\partial_\theta$$ be a basis of your distribution. Horizontal curves are trajectories $z :[0,T]\to M$ (here $M=\mathbb{R}^2\times S^1$), that are tangent to the distribution, that is there exist $u_1,u_2:[0,T]\to \mathbb{R}$ (let us forget about regularity issues), such that $$ \dot z(t) = u_1(t) X_1|_{\gamma(t)} + u_2(t)X_2|_{\gamma(t)}, \qquad t\in [0,1]. $$ You look for horizontal trajectories between fixed endpoints, that minimize the cost functional: $$ \int L(\gamma(t))dt = \frac{1}{2}\int_0^T \left(u_1(t)^2 + u_2(t)^2\right)dt. $$ (this should clarify the normalization of $X_1,X_2$).

This is a very simple optimal control problem. An application of the Pontryagin Maximum Principle yields first order necessary conditions for solutions of this problem. In this very simple all solutions correspond to projections on $M$ of integral trajectories of an Hamiltonian system on $T^*M$, with Hamiltonian given by

$$ H(\lambda) = \frac{1}{2}\sum_{i=1}^2 \lambda(X_i)^2, \qquad \lambda \in T^*M. $$

In canonical coordinates $(x,y,\theta,p_x,p_y,p_\theta)$ on $T^*M$, your Hamiltonian reads explicitly

$$ H=\frac{1}{2}(p_x \cos\theta + p_y \sin\theta)^2 + \frac{1}{2}p_\theta^2. $$

Then your "geodesic equations" correspond to the Hamilton's equations for $H$.

A good reference to this general kind of optimal control problems can be found in the following book (chapter 12, Theorem 12.10, case (b) is vacuous in your case, so only case (a) remains):

Agrachev, Andrei A.; Sachkov, Yuri L., Control theory from the geometric viewpoint., Encyclopaedia of Mathematical Sciences 87. Control Theory and Optimization II. Berlin: Springer (ISBN 3-540-21019-9/hbk). xiv, 412 p. (2004). ZBL1062.93001.

For a specifically sub-Riemannian reference, see Chapter 4.3 in

Agrachev, Andrei; Barilari, Davide; Boscain, Ugo, A comprehensive introduction to sub-Riemannian geometry. From the Hamiltonian viewpoint., ZBL07073879.

The references proposed by Robert Bryant are also very good, but the two aforementioned one are closer (I think) to the spirit of your question.

Answer 2

There are a number of sources. If you are comfortable using differential forms, then one source that you may find useful is

L. Hsu, Calculus of variations via the Griffiths formalism, J. Differential Geom. 36, 551–589 (1992).

He explains how to do this calculation very explicitly there.

Another source is

R. Montgomery, A Tour of Subriemannian Geometries, Their Geodesics and Applications, AMS Mathematical Surveys and Monographs, Volume 91. 2002. MathSciNet review: MR1867362

You can get a pdf download from the American Mathematical Societ web page https://www.ams.org/books/surv/091/